The \(k\)-Buchsbaum property for some polynomial ideals
Let \(R\) be a standard graded ring over a field \(K\), and let \(\mathfrak m\) be the maximal homogeneous ideal. A graded module \(M\) over \(R\) is said to be \(k\)-Buchsbaum if its local cohomology modules \(H^i_{\mathfrak m} (M)\), \(0 \leq i \leq d\), are annihilated by \({\mathfrak m}^k\), where \(d+1\) is the Krull dimension of \(M\). In this article, the authors focus on the case where \(R = R_{r+1} := K[x_0,\dots,x_r]\) and \(M = R/\mathfrak a\), where \(\mathfrak a\) is a homogeneous ideal and \(K\) is infinite. An equivalent (and, according to the authors, more workable) definition for \(k\)-Buchsbaum ideals, and some technical reformulations, is given in terms of systems of parameters. Much earlier, \textit{H. Bresinsky} and \textit{W. Vogel} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 39, 143--159 (1993; Zbl 0837.13008)] had given an algorithm to test if such an ideal is Cohen-Macaulay or Buchsbaum. (The former corresponds to \(k=0\); the latter means that \(R_{r+1}/{\mathfrak a}\) and \(R_{r+1}/({\mathfrak a},F_0,\dots,F_i )\), \(0 \leq i \leq d\), are 1-Buchsbaum for any system of parameters.) The central problem addressed in this paper is to find an algorithm to determine, or at least bound from above, the \(k\)-Buchsbaumness of an ideal, for larger \(k\), without explicit computation of Ext-modules or local cohomology modules. The authors are successful in the case of certain binomial ideals, using Gröbner bases computations, and in fact they find the smallest such \(k\).
scientific article; zbMATH DE number 2124474